Long-range interactions
for linearly elastic media resulting in
nonlinear dispersion relations are modelled by an
initial-value problem for an integro-differential equation (IDE) that
incorporates non-local effects. Interpreting this IDE as an
evolutionary equation of second order, well-posedness in
$L^{\infty}(\rz)$ as well as jump
relations are proved. A numerical approximation based upon quadrature
is suggested and carried out for two examples, one
involving jump discontinuities in the initial data corresponding to a
Riemann-like problem.